The upgrade would be with the same network, but if you spot a deal offering similar minutes, texts, and data from another network (and have easy way to backup all contents from your mobile) you could get current network to give you a PAC (Porting authority code) and move your number to a different network.
You'd get a monthly contract via CPW for the new deal, including the new phone, and then move your number (using that PAC within 30 days from when it is issued). When you use the PAC, the old phone will no longer have service from the old network, and your new phone will magically get the old number (the one you are using now) moved onto it, so friends will still be abele to reach you.
The other possibility for an "upgrade" price on CPW site is for a new PAYG phone, when you buy the phone from CPW. Buying a phone gets a price which may depend on the network. If you've been spending regularly with a network, the price charged for a PAYG phone is sometimes reduced (but not by much really), although with an iPhone 6 you'd be looking at hundreds of pounds and I'm guessing you want to continue under contract and upgrade to the iPhone 6 while not spending any extra...
What are you currently paying, and what prices (for upgrade and monthly contract) have you seen ? (Oh, and which network are you on, for complete clarity?) Might be better to see some examples of what you've seen and then hopefully be able to explain why there are differences :)